The present invention relates in general to communication systems and methods, particularly those employing chaotically varying transmission signals, which incorporate receivers using estimators, such as extended Kalman filters, for estimating the states and parameters in corresponding transmitters to facilitate synchronization therewith and reception of signals therefrom.
Being able to synchronize two or more remote systems has applications in communications, control and similar fields. Once synchronized, such systems lend themselves to various communications techniques. Current methods are based on linearity, or regular nonlinear behavior, or require that stability restrictions be imposed upon subsystems.
Nonlinear systems can possess more than a single equilibrium point; this property can give rise to more complex dynamics than is generally observed in linear systems, which possess only a single equilibrium point. Nonlinear systems can exhibit stable, unstable, cyclic or chaotic behavior, and it is entirely possible that a single nonlinear system can exhibit all four types of behavior depending upon the choice of operational parameters and/or initial conditions. It is also possible that almost every nonlinear system has some range of parameters and/or set of initial conditions for which it will exhibit chaotic behavior.
Chaotic behavior can be characterized by a sensitivity to initial conditions, that is, two trajectories starting from arbitrarily close, but different, initial conditions will always diverge from each other as time passes, and eventually become completely uncorrelated, although they both continue to exhibit the same characteristics of behavior. Chaotic systems are deterministic systems that exhibit random appearing behavior.
Synchronization of two chaotic systems has been disclosed, for example, in U.S. Pat. Nos. 5,245,660 and 5,379,346 to Pecora, et al., and in U.S. Pat. No. 5,473,694 to Carrol, et al., however, their methods are restricted to only those systems that can be decomposed into stable subsystems.
In U.S. Pat. No. 5,291,555, Cuomo, et al., two methods of transmission are disclosed: the first method is to add a signal to the chaotic transmitter output, which is then transmitted as a sum of the two wave forms. The signal is recovered at the receiver by subtracting the synchronized chaotic carrier from the received wave form. The second method is to vary a parameter in the transmitter, causing the receiver to lose synchronization lock temporarily, and then regaining lock when the parameter is toggled back, creating a binary bit stream of lock-unlock which is decoded by measuring the energy in the error signal.
The method of adding a message to the chaotic carrier is very sensitive with respect to the signal to carrier to noise power ratios, making it difficult to send a message with low enough power to avoid detection, yet strong enough to be heard above noise. The second method of locking-unlocking for bit transmission is also very sensitive with respect to additive noise, since the noise appears directly in the error signal.
In 1960, Richard E. Kalman published a paper entitled "A New Approach to Linear Filtering and Prediction Problems" (Journal of Basic Engineering 82D, pp. 35-45, 1960) in which he disclosed a linear filtering technique that can be employed for estimating the states of a system based upon initial state conditions and measurements of the system over time. Devices constructed in accordance with this technique became known as Kalman filters. Subsequently, this concept was applied to estimation of a system's parameters in addition to its states, and the device for accomplishing this became known as the extended Kalman filter (EKF).
EKFs have been used in a computer simulation to estimate the states of a chaotic system. However, to the inventors' knowledge, EKFs have never been used in communication systems for estimating the states and parameters of a transmitter and thereby facilitating synchronization of a receiver with the transmitter.